I've donated blood several times and I never look when they stick me with the needle. But that's because I'm a baby when it comes to needles and I'd pull away if I saw someone approaching me with it.
But once, I went to the hospital with a friend and they had to draw blood from her. She's worse than me when it comes to needles so I stayed there to comfort her. I watched as the nurse as went to draw blood and there was a messy moment when there was just too much blood coming out of my friend's arm. Normally the sight of blood doesn't bother me, but the way it was coming out of her arm really bothered me, even though it was just a couple of seconds.
If I had to witness something like that, from my own arm, I would freak out. I'd probably end up causing a scene, passing out or just causing damage to myself. Perhaps, all three at once. And maybe that's why they advise you not to look.
I hope that makes sense . . .